I think I agree with your second statement:
B_{li}=\sum_{j}A_{lj}A_{ij}
In my books notation that would be:
\sum_{ilj}A_{lj}A_{ij}e_{i}e_{l}=\sum_{ilj}A_{ij}A_{lj}e_{i}e_{l}=A \cdot A^{T}
Since A_{ij} is just a constant it can be rearranged.
correct?
While I do not disagree with you, and using your method does seem to give me the correct answer my book explicitly states that given a tensor \tau:
\tau=\sum_{ij}\tau_{ij}e_{i}e_{j}
then \tau^{T}=\sum_{ij}\tau_{ji}e_{i}e_{j}
I am assuming this means that the transpose would be when you only...
correct, based off my notation ei and ej represent the tensors and Aij represents the coeffiecent of the tensor. The problem lies I feel in what the coefficent should be when you take the transpose. I get AljAij as my transpose coeffiecent, but it should be equivalent to AijAlj, and I am not...
My understanding for my transport book is that coefficient is effected by transposition, which is what I showed. The notation I am using corresponds pretty similarly to what is presented here: http://www.foamcfd.org/Nabla/guides/ProgrammersGuidese3.html
With the e's representing the tensor...
I guess I will clear up some of my notation since I don't think a Dirac delta comes into place here. The e in this case represents the unit dydas, and delta is a standard Kronecker delta.
So form the start:
\bar{}
\sum_{ij} A_{ij}e_{i}e_{j}\cdot...
the deltas represent the unit dydas (according to the book I am using) and I agree with your matrix product however the other A is transpose so I would think it would be the sum over A_ij*A_kj which is essentially what I wrote in my original post.
Homework Statement
Show that A \cdot A^{T} is symmetric (A is a 2nd order tensor)
Homework Equations
The Attempt at a Solution
So I got down to this and i can see that it will be symmetric however when I try taking the transpose of the solution I can't seem to make it equal...
So the only thing I can think of is the pressure of the top liquid acting on the bottom liquid, as well as the buoyancy of the bottom liquid with respect to the top liquid.
Im assuming these two forces are equal to each other Fb= (h1+h0)A*p1. If I include the buoyancy of liquid 2 on liquid 1...
Homework Statement
A box with a height L, and a cross sectional Area A is floating between two fluids of densities p1 and p2. Determine the mass of the box using ho, h1, h2, p1, p2, and A.
Homework Equations
F= pVg
The Attempt at a Solution
In the file attached I drew my free...
Homework Statement
You've been given a pulley for your birthday. It is a fairly big pulley 12 cm in diameter and with mass 2kg. You get to wondering if the pulley is uniform. To find out, you hang a 1kg textbook 1.0 meters above the floor and use a stopwatch to measure the time it takes it to...